Hydrodynamic instabilities of a viscous liquid film flowing down an inclined or vertical plane

Abstract

In this paper, we have investigated theoretically linear as well as weakly nonlinear stability of a viscous liquid film flowing down an inclined or vertical plane under the action of gravity. The classical momentum-integral method, which is applicable for small as well as large values of Reynolds number Re, has been used to formulate the single nonlinear free surface equation in terms of the dimensionless perturbed film thickness η(x,t). Using sinusoidal perturbation in the linearized part of the surface evolution equation, we obtain the stability criterion and the critical value of the wave number kc which conceives the physical parameters Re, inclination angle θ and Weber number We. However, the linear stability analysis reveals the stabilizing influence of We as well as the destabilizing influence of Re and θ on this flow dynamics. The multiple-scale analysis has been used to derive the complex Ginzburg–Landau type nonlinear equation for investigating the weakly nonlinear stability analysis. We demarcate all the four states of the flow in the Re-k (or θ-k)-plane which are found after the critical value of Rec (or θc) depending upon the values of the other parameters. A novel result which emerges from the nonlinear stability analysis is a simple relationship among the parameters k, Re, We, and θ. This relationship essentially gives us the conditions needed for the existence of an explosive unstable zone when (3Re−3cot θ−4ReWek2)= 0; otherwise, the flow system will be free from this zone. Indeed, this zone decreases with the increase in We, whereas it increases with the increase in Re and θ confirming the stabilizing role of We and destabilizing role of Re and θ as found in linear stability analysis.

Electromagnetic field effect on a conducting liquid film flowing down an inclined or vertical plane

“…For , whatever may be the value of , (4.4) takes the form , which is exactly the same as reported by Alekseenko et al. (1994) and Dholey & Gorai (2021).…”

“…Integrating (2.12) and then using (2.16), we obtain the dimensionless pressure as Integrating the continuity equation (2.10) and the -momentum equation (2.11), after using (3.1), with respect to from to by the Leibnitz rule, and using the boundary conditions (2.13 a , b )–(2.15), we have where the expressions of the shape factors and are given by Equation (3.4) confirms that the shape factors and depend only on the values of . For , , which are exactly the same as the values obtained by Dholey & Gorai (2021). The full range of can be obtained as since as .…”

“…The nonlinear evolution equation (4.5) of our present study may be the same as the corresponding equation in Mukhopadhyay et al. (2008), but the results obtained from figures 3–8 of their analysis are not correct, as the curves and (in those figures) do not follow the known relation (Dholey & Gorai 2021). Dandapat & Mukhopadhyay (2003) have shown the existence of two critical values of , namely, and , in figure 3 of their analysis.…”

“…It is noticeable that all the above studies are restricted to a definite value of or in a specific range of in which is small. Recently, using the momentum-integral method, Dholey & Gorai (2021) examined thoroughly the linear and nonlinear stability analysis of a viscous liquid film falling down an inclined plane for the full range of the inclination angle ().…”

“…The aim of this study is therefore to extend the work of Dholey & Gorai (2021) by considering the flow of an electrically conducting liquid film down an inclined plane in the presence of an electromagnetic field. The linear stability analysis reveals that the magnetic parameter has a stabilizing influence up to the value , independent of , and after this value of , the (slowly) destabilizing influence of has been found up to the value , and then continuously follows the stabilizing influence.…”